The Curious Case of Integrator Reach Sets, Part I: Basic Theory

Abstract

This is the first of a two part paper investigating the geometry of the integrator reach sets, and the applications thereof. In this Part I, assuming box-valued input uncertainties, we establish that this compact convex reach set is semialgebraic, translated zonoid, and not a spectrahedron. We derive the parametric as well as the implicit representation of the boundary of this reach set. We also deduce the closed form formula for the volume and diameter of this set, and discuss their scaling with state dimension and time. We point out that these results may be utilized in benchmarking the performance of the reach set over-approximation algorithms.

Certifying the Intersection of Reach Sets of Integrator Agents With Set-Valued Input Uncertainties

“…This is because the input correspondence (7) occurs via time-varying x. Thus, we cannot apply recent results [9]- [11] explicitly characterizing the integrator reach set boundary with time-invariant input range.…”

“…Thanks to the Lyapunov convexity theorem [15]- [18], the reach sets (10) and ( 11) are guaranteed to be compact, convex [19,Prop. 6.1], [3], and in particular, zonoids 1 [9]- [11].…”

“…We propose to use the input correspondence (7) for explicitly computing (11) in two steps: step 1: explicitly compute the (boundary of the compact) integrator reach set X t ({M z 0 }) for to-be-determined input range [u min (s), u max (s)] ∀s ∈ [0, t]. step 2: compute Z t ({z 0 }) = M −1 X t ({M z 0 }).…”

“…Furthermore, the LTI reach set (11) with time-invariant input range [v min , v max ] can be recovered from the integrator reach set (10) with time-varying input range [u min (s), u max (s)] given by (12), as…”

“…• We show (Sec. II) how certain generalizations of recent results [9]- [11] on the exact geometry (e.g., boundary, volume) of integrator reach sets can enable computing the controllable LTI reach sets with bounded input. Our results reveal that the controllable canonical form serves as a bridge to transfer such geometric results from the integrator to the original state coordinates.…”